But what I would like to find is the instantaneous position and direction after a given time, say 1 nanosecond.

I tried dividing the circumference by 131 nanoseconds, but I feel like this is cheating.

I also tried to use

Acceleration = Force/mass, but in this case the acceleration is only a change in direction, not in velocity.

It is no cheating if you calculate the angular position as
alpha = 2pi*(t/T). And you know that the proton moves along a circle with constant speed, so the velocity vector is tangent to the circle. Knowing alpha, you also get the direction of the velocity.